is the bipolar spherical harmonics [1]} for a
pair of unit vectors \({n}_1\)  and \({n}_2\) , and the coefficient
\(c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)\)  is the invariant function of the
second order. The indices \((l,l_1,l_2)\)  of the coefficient
\(c^{(2)\,l}_{Xl_1l_2}(k_1,k_2)\)  satisfy a triangular condition,
\(|l_1 - l_2| \ge l \ge l_1+l_2\)  as those in the bipolar spherical
harmonics of Eq. (REF ). While the second-order function
\(c^{(2)}_{Xlm}({k}_1,{k}_2)\)  is invariant under the interchange
of \({k}_1 \leftrightarrow {k}_2\) , the invariant function is not
invariant under the interchange of the subscripts 1 and 2. Instead,
due to the symmetry of the \(3j\) -symbol,
\((l\,l_1\,l_2)_{mm_1m_2} = (-1)^{l+l_1+l_2}(l\,l_2\,l_1)_{mm_2m_1}\) ,
we have an interchange symmetry,
\(c^{(2)\,l}_{Xl_2l_1}(k_2,k_1)= (-1)^{l+l_1+l_2}c^{(2)\,l}_{Xl_1l_2}(k_1,k_2).\) 
